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Flame-sheet dynamics of bluff-body stabilized flames during longitudinal acoustic forcing
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Combustion Institute
Proceedings of the Combustion Institute 32 (2009) 1787–1794
www.elsevier.com/locate/proci
Flame sheet dynamics of bluff-body stabilized flames during longitudinal acoustic forcing Santosh Shanbhogue *, Dong-Hyuk Shin, Santosh Hemchandra, Dmitriy Plaks, Tim Lieuwen Ben T. Zinn Combustion Laboratory, Georgia Institute of Technology, 635 Strong Steet, Atlanta, GA 30318, USA
Abstract Bluff-body stabilized flames are susceptible to combustion instabilities due to interactions between acoustics, vortical disturbances, and the flame. In order to elucidate these flow-flame interactions during an instability, an experimental and computational investigation of the flame-sheet dynamics of a harmonically excited flame was performed. It is shown that the flame dynamics are controlled by three key processes: excitation of shear layer instabilities by the axially oscillating flow, anchoring of the flame at the bluff body, and the kinematic response of the flame to this forcing. The near-field flame features are controlled by flame anchoring and the far-field by kinematic restoration. In the near-field, the flame response grows with downstream distance due to flame anchoring, which prevents significant flame movement near the attachment point. Theory predicts that this results in linear flame response characteristics as a function of perturbation amplitude, and a monotonic growth in magnitude of the flame-sheet fluctuations near the stabilization point, consistent with the experimental data. Farther downstream, the flame response reaches a maximum and then decays due to the dissipation of the vortical disturbances and action of flame propagation normal to itself, which acts to smooth out the wrinkles generated by the harmonic flow forcing. This behavior is strongly non-linear, resulting in significant variation in far-field flame-sheet response with perturbation amplitude. Ó 2009 The Combustion Institute. Published by Elsevier Inc. All rights reserved. Keywords: Combustion instabilities; Bluff-body flames; Flame dynamics; Flame kinematics
1. Introduction The objective of this paper is to describe the flame dynamics of acoustically forced, bluff-body flames. The motivation for this work is 2-fold.
* Corresponding author. Address: Reacting Gas Dynamics Laboratory, 3-339 Mechanical Engineering, 77 Massachusetts Avenue, Massachusetts Institute of Technology, Cambridge, MA 02139, USA. Fax: +1 617 253 5981. E-mail address: [email protected] (S.J. Shanbhogue).
First, the dynamics of such flames, even in the absence of imposed excitation, involves complex interactions between gas expansion effects, flow instabilities, and vorticity dynamics induced by both shear and combustion, as highlighted by, e.g., Poinsot and Veynante [1], Schadow et al. [2] and Rogers and Marble [3]. As such, this is a fundamental problem that involves complex interactions between a variety of fluid mechanic and combustion processes [4,5]. In addition, it is motivated by the problem of combustion instabilities in devices utilizing these types of flame-holders, such as afterburners [3]. In many such instances,
1540-7489/$ - see front matter Ó 2009 The Combustion Institute. Published by Elsevier Inc. All rights reserved. doi:10.1016/j.proci.2008.06.034
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it is known that vortical structures generated from these flame-holders interact with the flame downstream, causing its heat release to oscillate [6]. In the non-reacting case, the flow-field dynamics are dominated by that of the free shear layer and wake, each exhibiting distinct instabilities with their respective frequency/amplitude response characteristics [7]. For the conditions explored in this paper, the instability of the free shear layer is of particular significance [3,8]. The shear layer (or Kelvin–Helmholtz, KH) instability is a convective instability leading to vortex roll-up and pairing. Under the influence of harmonic excitation, the separated shear layer rolls up into vortices with a frequency equal to that of the imposed excitation [9]. In addition, they lead to velocity fluctuations observed at sum and difference frequencies of the forcing frequency and its harmonics, due to non-linear interactions [9]. In the presence of heat release, the flow dynamics are fundamentally altered. For example, studies of the impact of heat release on planar shear layers indicate reduced vortex structure growth rates and Reynolds stresses compared to the non-reacting situation [10,11]. For premixed bluff-body flames, the most prominent effect is the suppression of the absolute wake instability, which leads to the von-Ka´rma´n vortex street, for flames with density ratios greater than 2 [12]. This is the reason that the shear layer instability dominates the flame response characteristics highlighted in this paper. A number of prior studies have characterized the interaction of flames with harmonic waves arising due to both acoustic waves [13] and convecting, vortical disturbances [14–18]. The dynamics of the flame are controlled by flame kinematics, i.e., the propagation of the flame normal to itself at the local burning velocity, and the flow-field that the flame is locally propagating into. This is mathematically described by the G-equation [1,19]: oG þ~ u rG ¼ S L jrGj ot
ð1Þ
where the flame position is described by the parametric equation Gð~ x; tÞ ¼ 0. Also, ~ u ¼~ uð~ x; tÞ and SL denote the flow-field just upstream of the flame and laminar burning velocity, respectively. In the unsteady case, the flame is being continually wrinkled by the unsteady flow-field, ~ u0 . The action of flame propagation normal to itself, the term on the right side of Eq. (1), is to smooth these wrinkles out through ‘‘Huygens propagation”/‘‘kinematic restoration”. As such, a wrinkle created at one point of the flame due to a velocity perturbation is convected downstream and diminishes in size due to kinematic restoration. Indeed, the interaction between the excitation (acoustic/vortical flow oscillations) and the damping (restoration property of the flame) can lead to a range of effects depending upon flame stabilization and the
relative values of the flow oscillations and flame speed. This manifests itself through both local influences upon the flame topology [20] (e.g., cusping, amplitude of corrugation, pocket formation), and global influences upon the overall unsteady heat release response of the flame. The goal of the present work is to elucidate the flame-sheet dynamics that results from an acoustically forced bluff-body flame. First, we experimentally study the dynamics of the perturbed bluff-body flame, i.e., the spectral and spatial characteristics of the flame position and the effects of excitation amplitude and frequency. Then, we show that the key characteristics of these response curves can be understood from an analysis of the G-equation. These theoretical solutions show that the near-field features are controlled by flame anchoring and the far-field by kinematic restoration. 2. Experimental facility and approach Experiments were carried out in a 915 mm long atmospheric pressure burner with a square crosssection (95 mm 95 mm), shown in Fig. 1a. Fuel, air and fine (1–5 lm) olive oil are introduced in a mixing chamber located at the base of the burner. The mixture exits the mixing chamber into a 150 mm tube having the same cross-section as the burner, which houses two 100 W acoustic loudspeakers. The mixture then passes through an aluminum honeycomb flow straightening section. The bluff body is mounted at the immediate exit of the channel. The flame is visualized with a laser sheet that scatters light off the olive oil droplets in the reactants. A 1 Watt Coherent Innova 70-5 Ar+ laser was used for illumination. A cylindrical lens arrangement was setup to provide a 2 mm light sheet that passes through the flame at the mid point of the bluff body. The resulting field of view spanned 109 mm 109 mm and was imaged using a Phantom Ultracam3 intensified camera with a 514 nm laser line filter and a BG-7 Schott Glass absorption filter placed in front of a Nikon 55 mm, f#-2.8 lens. At each operating point and/ or excitation amplitude, 2048–4096 images were obtained at 500–1000 frames/s with an exposure time of 400 ls. The acoustic drivers were sinusoidally forced with a function generator, connected to an amplifier. Typical axial and transverse cuts of the flame sheet are shown in Fig. 1b at an amplitude of excitation where the transverse flame location is multi-valued. The vertical image is in the direction of the mean flow and the horizontal image is a transverse cut obtained by rotating the laser sheet by 90°. Velocity measurements were obtained using phase locked particle image velocimetry (PIV) using a dual head Nd:YAG laser of wavelength 532 nm and peak power output of 120 mJ/pulse.
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Fig. 1. (a) Schematic of the experimental facility; dimensions are in millimeters. (b) Instantaneous axial (obtained at bluff-body centerpoint) and span-wise (obtained at axial location y/D = 3) Mie scattering images (conditions: Uo = 1.8 m/s, fo = 130 Hz, u0 /Uo = 0.06, / = 0.8, circular bluff body).
A 1600 1200 pixels CCD camera, with an Fmount Nikon 55 mm micro-lens with an aperture of f/5.6 was used for imaging. The interval between the beam pulses was set to 30 ls. The light sheet was generated using two cylindrical lenses of 150 and 1000 mm focal length that reduced the thickness of the beam and caused the 5 mm laser beam to diverge to a height of 40 mm. The distance between the imaging plane and the camera was set at 30.5 cm. Seed consisted of 1.5 lm Al2O3 injected into the setup with a cyclone seeder. At each phase, 128 images were recorded and the resulting data was ensemble averaged to provide velocity uncertainties of approximately 2%. The field of view was split into interrogation regions of 64 64 pixels, with a 50% overlap, yielding a spatial resolution of 1.5 mm. Mean velocities (Uo) and acoustic velocity amplitudes, u0a , reported below were obtained with a calibrated hotwire anemometer at the jet exit without the bluff body mounted. Transverse mapping of the velocity field at the tube exit showed that the mean velocity and incident acoustic wave profile was planar to within 3% and 4%, respectively. Because of excitation of vorticity at the bluff body, the actual velocity disturbance exciting the flame is different from u0a . All experiments were conducted with natural gas as fuel. Triangular and circular bluff bodies with various widths, d, were used for flame stabilization. 3. Results A comparison of the flame and flow-field dynamics during forced and unforced conditions
is shown in Fig. 2, which overlays instantaneous and ensemble averaged vorticity contours with instantaneous flame-sheet locations. In the absence of acoustic forcing (Fig. 2a), i.e., when u0a =U o ¼ 0, no periodic activity is present in the flame-sheet fluctuations, as determined from spectral analysis of the flame-sheet position [21]. On an instantaneous basis, random, roughly symmetric corrugations on the flame are evident leading to a flame brush growing in thickness with downstream distance, similar to the observations of Be´dat and Cheng [22] and Knaus and Gouldin [4]. As acoustic forcing is introduced (Fig. 2b–d) the presences of symmetric distortions are evident on the flame, whose amplitudes are dependent on the excitation magnitude. At larger distances from the bluff body, particularly at higher amplitudes, cusping of the flame front can also be seen, indicative of the action of the non-linear kinematic restoration process due to flame propagation normal to itself, see Law and Sung [23]. At even higher amplitudes (Fig. 2d) the complete roll-up of the flame by the convecting vortical structure is also seen. The period of formation of these structures is identical to that of the acoustic excitation, fo. The vortical structures originate in the shear layer of the bluff body and propagate in the direction of the flow. These images also show that the vorticity/velocity field that perturbs the flame decays relatively quickly, and that the flame-sheet corrugations persist further downstream. These observations can be quantified by tracking the flame edge. The coordinate system (see Fig. 2b) is chosen such that the y-axis points in the direction of the mean flow. The (x, y) = (0, 0) points is on the downstream edge of the bluff body, on the centerline. The flame
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Fig. 2. Instantaneous images of the flame-sheet and vorticity field (a) without excitation and (b–d) with increasing amplitudes of excitation (d = 9.52 mm, Uo = 2.7 m/s, fo = 300 Hz and / = 0.72).
location, L(y,t) is determined at each time instant, t, and each spatial location, y for conditions where the flame sheet is a single valued function of y (i.e., lower u0a values, no quantitative data is shown here for flames with multi-valued L(y,t)). The mean value of L at each y location is subtracted to obtain the flame-front perturbation value L0 ðy; tÞ ¼ Lðy; tÞ LðyÞ. This signal is then decomposed into its spectral components using the Fourier transform to yield L0 (y,f).
Typical characteristics of flame-front position spectra (amplitude and phase), under the influence of acoustic excitation are shown in Fig. 3. The convective wavelength of the flame-front disturbances, kcf = Uo/fo, equals the distance a disturbance propagating at the mean flow velocity travels in one acoustic period. The plot on the left shows the spectrum of the flame response at seven downstream locations. The envelope of the flame response at f = fo is also drawn. Close to the bluff
Fig. 3. (a) Dependence of flame-sheet fluctuation spectrum, L0 (y,f) upon axial location (Uo = 4.5 m/s, fo = 300 Hz). (b) Phase dependence upon normalized axial location, where yo indicates first axial location where data was obtained (d = 9.52 mm, u0a =U o ¼ 0:05 and / = 0.67, triangular bluff body).
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body, the flame responds chiefly at the frequency of excitation (fo). Moving downstream, the response at f = fo first grows, reaches a maximum, and then decreases. This behavior is due to the growth and decay of the underlying flow structures as well as the propagation of the flame, which tends to smooth out the wrinkles. These results are consistent with other data, e.g., Hegde et al. [24], though they have apparently not been explicitly discussed. The spectrum also exhibits a monotonic increase in broadband fluctuations with downstream distance. This reflects the random flapping of the flame brush, which increases in magnitude with downstream axial distance, as is clearly seen for the unforced case, see Fig. 2a. The corresponding axial variation of the phase at the excitation frequency is plotted in Fig. 3b. For all these conditions, the phase behavior nearly collapses onto a single line when axial distance is normalized by kc, indicating a constant axial convection speed of these flamesheet disturbances of Ucf = 0.98Uo. In general, there is a mild dependence of the convection speed on the equivalence ratio and the bluff-body shape [21], ranging between 0.8 < Ucf/Uo < 1.1. Although not shown, this convection speed is nearly independent of excitation amplitudes, at least for amplitudes below where the flame becomes multi-valued [21]. Consider the amplitude characteristics in more detail. Figure 4a plots the dependence of the amplitude upon normalized axial distance at three excitation amplitudes, but all other conditions remaining equal. Figure 4b plots these same data, with the y-axis normalized by the acoustic excitation amplitude. The plot can be divided into two regions manifested by ‘‘growth” and ‘‘decay” of the flame-sheet oscillations. This plot shows that the flame response increases linearly with excitation amplitude in the bluff-body near-field, or ‘‘growth” region, as manifested by all the curves
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in Fig. 4b converging onto a single line. However, farther downstream, the flame dynamics are clearly non-linear, as again evidenced by the amplitude normalized response plots diverging with increasing axial distance. In particular, the saturation in flame response can be seen, indicating that the maximum in flame response, as well as response farther downstream, does not grow proportionally with amplitude. As shown next, this is due to the non-linearities associated with the kinematic restoration process, which plays the dominant role in this region. In contrast, in the growth region, the flame dynamics are linear and primarily influenced by the flame-anchoring condition near the stabilization point. In order to better understand these features, the next section describes computational analyses of this configuration. 4. Kinematic modeling of flame-sheet dynamics In this section, we consider the role of flame kinematic processes in the basic flame response characteristics described above. As will be shown, the flame dynamics are controlled by three processes: (1) the anchoring of the flame at the bluff body, (2) the excitation of wrinkles by the oscillating velocity, and (3) flame propagation normal to itself at the local flame speed, smoothing out the wrinkles introduced by the oscillating flow. Utilizing the coordinate system defined in Fig. 2b, we write the function G in Eq. (1) as G(x,y,t) = 0 = L(y,t) x, as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi oL oL oL v þ u ¼ SL 1 þ : ð2Þ ot oy oy This equation is solved with a flame-anchoring boundary condition at y = 0; i.e., whether and to what extent the flame base moves in response to excitation. Analytical solutions of
Fig. 4. (a) Measured dependence of flame sheet amplitude response upon normalized axial distance, kc = Uo/f. (b) Same curve, but amplitude normalized by acoustic velocity amplitude: () u0a ¼ 0:028, (h) u0a ¼ 0:021, (+) u0a ¼ 0:016, (/) u0a ¼ 0:010. Other conditions: Uo = 2.27 m/s, D = 12.7 mm, fo = 150 Hz, cylindrical bluff body.
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the linearized version of this equation are possible for low amplitudes and/or small y/kc values, but it requires numerical solutions for the general non-linear case. A fifth-order Weighted Essentially Non-Oscillatory (WENO) numerical scheme [25] was used, uniformly fifth-order accurate in space in the smooth regions and thirdorder accurate in discontinuous regions. Derivatives at the boundary nodes were calculated using third-order accurate upwind-differencing schemes so that only the nodes inside the computational domain are utilized. A third-order accurate Total Variation Diminishing (TVD) Runge– Kutta scheme [26] is used for time integration. The computations were performed for a grid of length 6kc with a grid point spacing Dy = 0.001kc. Sensitivity studies performed with a grid 10 times finer demonstrated that the difference between the two grid density results was less than 0.1%. The experimental data, e.g., Fig. 3a shows that the flame response first grows with downstream distance, up to an axial location of y/kc 0.9, before decaying. This initial increase in L0 with y is due to the flame anchoring; i.e., regardless of the perturbation, the flame attachment point remains largely fixed. As such, the amplitude of L0 must start from zero or near zero. Obviously, it will not remain zero at y > 0, since it is being perturbed by the fluctuating velocity field— hence the initial increase in L0 with y. Even if the flame attachment point vibrates some, i.e., L0 (y = 0,t) – 0, as long as the flame base does not move in phase and with the same amplitude as the flow-field, similar behavior occurs. Furthermore, because the amplitude of fluctuations in the y 0 region are so small, non-linear effects are negligible.rThese non-linear effects are contained ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi in the S L 1 þ oL term, which describes flame oy propagation normal to itself. As such, the flame dynamics near the attachment point are described by a linearized equation form of Eq. (2). An explicit solution for the slope of the jL0 (y,fo)j vs. y curve can be derived from Eq. (2) near the attachment point: ojL0 ðy ¼ 0; fo Þj ju0n ðy ¼ 0; fo Þj 1 ¼ oy U t ðy ¼ 0Þ cos2 h
ð3Þ
where u0n ; U t , and h denote the fluctuating velocity normal to the flame, mean velocity component tangential to the flame, and mean flame angle with respect the axial coordinate, see Fig. 2a. This equation assumes that L0 (y = 0,t) = 0 at y = 0. It can be seen that the near-field flame response slope increases linearly with perturbation magnitude, u0n . This result, together with the result in Fig. 4 suggests that the near-field flame response is controlled purely by the velocity field near the flame attachment point. This explains why the
curves in Fig. 4 converge to a common behavior at y 0. The peaking and subsequent reduction in amplitude of flame response shown in Fig. 4 is the result of flame propagation normal to itself which destroys flame wrinkles—a non-linear effect [23]. Moreover, for a thermo-diffusively stable flame, unsteady curvature effects also work to destroy flame wrinkles—even at first-order in perturbation amplitude—due to the increase in flame speed at locations concave to the flow and vice versa [18]. If the perturbation velocity persisted indefinitely downstream, at some point the flame wrinkle excitation and destruction processes would equilibrate, leading to a constant or oscillating amplitude of flame perturbation with downstream distance. However, the amplitude of the excitation field decays with downstream distance, as shown in Fig. 2. Hence, the amplitude of flame wrinkling decays. PIV velocity field measurements were obtained at one test condition to enable a comparison between predictions and calculations. Based upon the data, the transverse velocity field at the flame was fit to an equation of the form: vðy; tÞ ¼ vo ðyÞ þ ev y cosðxðt y=U c ÞÞecy :
ð4Þ
This equation shows that the perturbation is propagating axially with a velocity Uc and decaying at a rate given by c. The vortex convection velocity, Uc, is not necessarily equal to the flame-sheet convection velocity, Ucf, discussed in the context of Fig. 3. A similar expression, motivated by the continuity equation, was used for the axial velocity component, u0 (y,t) = e(1 + cy) ecycos(x(t y/Uc) w). These velocity field parameters were fit from the data, leading to c = 2.28/kc, w = 2p/3, Uc/Uo = 0.8, e = 1.4ev/c2kc (same conditions as in Fig. 3). A comparison of these velocity fits and the measurements are shown in Fig. 5. In addition, the frequency, fo, was determined from the known forcing frequency, h was determined from the flame images in the unforced case, and L0 (y = 0,t) was estimated as zero, as was observed in measurements (Fig. 4)—these quantities are also needed to solve Eq. (2). Direct comparisons of flame position dynamics were made in the near-field region only for two reasons. First, obtaining good velocity field data requires excitation at amplitudes that leads to the flame becoming multi-valued (e.g., see Fig. 2d) downstream. A further complication, particularly with velocity field specification further downstream, occurs because of flame-sheet movement—measurements of the ensemble average perturbation velocity field just upstream of the flame front (i.e., at a temporally varying location), hu(x(t),y(t))i, or at a spatially fixed point coinciding with some ‘‘average” flame-front location, hu(xo,yo)i, give different results. Both results
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Fig. 5. Comparison between the model and measurements of perturbation velocity amplitudes upstream of flame surface. Varying point: velocity measured at a location which is fixed relative to the moving flame. Fixed point: velocity measured at a fixed point in space. Conditions same as in Fig. 3.
are illustrated in Fig. 5 (compare ‘‘Fixed point” and ‘‘Varying point”). Nonetheless, very good quantitative comparisons are possible in the flame near-field, see Fig. 6. Starting first with the amplitude, inserting the measured velocity field data into Eq. (3), leads 0 o Þj 1 to a predicted near-field slope of junUðy;f ¼ cos2 h t ðyÞ 0:15 0:04=0:02 (using hu(x(t),y(t))i), in excellent 0 o Þj ¼ 0:16 agreement with the value of ojL ðy¼0;f oy 0:01 that was estimated from the flame-sheet fluctuation measurements. A corresponding comparison of the phase is also shown in Fig. 6—notice that their slopes are nearly identical, differing by about 2%. The two curves have a nearly constant offset for larger y values, due to a variation in the experimental phase slope near the attachment point, y ? 0. Computed solutions for several amplitudes and over a larger range of axial positions of flame
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Fig. 7. Predicted flame-sheet response based upon solution of non-linear flame-front equation at various perturbation amplitudes. For all cases c = 2.28/kc, w = 2p/3, Uc/Uo = 0.8, e = 1.4ev/c2kc. Other conditions same as in Fig. 3.
position amplitude are shown in Fig. 7. Notice that the behavior observed experimentally (Fig. 4a) is also seen here. First, the flame response is characterized by a near-field growth and a far-field decay region. Next, the flame response increases linearly with axial distance in the near-field region. Although not plotted, normalization of these curves by perturbation amplitude collapses them onto a single line, similar to the data in Fig. 4 (right), demonstrating the linearity in near-field flame response with amplitude as well. Next, Fig. 7 shows the divergence of these curves in the far-field region, demonstrating the far-field non-linear response. Finally, it captures the increased destruction rate of flame wrinkling, as manifested by the increased negative slope in the ‘‘decay” region, with perturbation amplitude. This latter point is illustrated by the growing difference between the linearized solution (scaled by perturbation amplitude) and the full non-linear solution with increasing y. As such, this shows the far-field decay characteristics of the flame surface amplitude are due to both velocity field decay and flame kinematic non-linearity.
5. Concluding remarks
Fig. 6. Comparison of axial dependence of flame-sheet amplitude and phase between experiments and theory. Conditions same as Fig. 3.
In this paper, two primary contributions have been made in understanding the response of bluff-body flames to harmonic excitation. First, through experiments, the coherent flame response was shown to have two different behaviors in the near and far-field of the bluff body, dominated by linear and non-linear flame kinematic processes, respectively. Second, using a kinematic model that encapsulated the dominant features of the perturbing velocity field; the key processes controlling the response have been
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identified as (1) the anchoring of the flame at the bluff body, (2) the excitation of flame-front wrinkles by the oscillating velocity, and (3) flame propagation normal to itself at the local flame speed. The first two processes control the growth of the flame response and the last process controls its decay. Acknowledgments This research was supported by the US-DoE and NSF under contracts DE-FG26-07NT43069 and CBET-0651045; contract monitors Rondle Harp and Dr. Phil Westmoreland, respectively. References [1] T. Poinsot, D. Veynante, Theoretical and Numerical Combustion, RT Edwards Inc., Flourtown, PA, 2001. [2] K.C. Schadow, E. Gutmark, T.P. Parr, D.M. Parr, K.J. Wilson, J.E. Crump, Combust. Sci. Technol. 64 (4) (1989) 167–186. [3] D.E. Rogers, F.E. Marble, Jet Propulsion 26 (1) (1956) 456–462. [4] D.A. Knaus, F.C. Gouldin, Proc. Combust. Inst. 28 (2000) 367–373. [5] F. Baillot, D. Durox, S. Ducruix, G. Searby, L. Boyer, Combust. Sci. Technol. 142 (1) (1999) 91– 109. [6] D.M. Kang, F.E.C. Culick, A. Ratner, Combust. Flame 151 (3) (2007) 412–425. [7] C.H.K. Williamson, Annu. Rev. Fluid Mech. 28 (1996) 477–539. [8] P.L. Blackshear, W.D. Rayle, L.K. Tower, Study of Screeching Combustion in a 6-in. Simulated Afterburner, Report No. NACA TN 3567, 1955. [9] A. Kourta, H.C. Boisson, P. Chassaing, H.H. Minh, J. Fluid Mech. 181 (1987) 141–161.
[10] P.A. Mcmurtry, J.J. Riley, R.W. Metcalfe, J. Fluid Mech. 199 (1989) 297–332. [11] J.C. Hermanson, P.E. Dimotakis, J. Fluid Mech. 199 (1989) 333–375. [12] R.R. Erickson, M.C. Soteriou, P.G. Mehta, The Influence of Temperature Ratio on the Dynamics of Bluff Body Stabilized Flames, AIAA Paper No. 2006-753, Presented at 44th AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada, 2006. [13] T. Lieuwen, Proc. Combust. Inst. 30 (2) (2005) 1725–1732. [14] A. Chaparro, E. Landry, B.M. Cetegen, Combust. Flame 145 (1–2) (2006) 290–299. [15] R. Balachandran, B.O. Ayoola, C.F. Kaminski, A.P. Dowling, E. Mastorakos, Combust. Flame 143 (1–2) (2005) 37–55. [16] C. Nottin, R. Knikker, M. Boger, D. Veynante, Proc. Combust. Inst. 28 (2000) 67–73. [17] T. Schuller, S. Ducruix, D. Durox, S. Candel, Proc. Combust. Inst. 29 (1) (2002) 107–113. [18] Preetham, T. Lieuwen, Nonlinear Flame-Flow Transfer Function Calculations: Flow Disturbance Celerity Effects, AIAA Paper No. 2004-4035, Presented at the 40th AIAA/ASME/SAE Joint Propulsion Conference, 2004. [19] V. Yang, F.E.C. Culick, Combust. Sci. Technol. 45 (1) (1986) 1–25. [20] A. Bourehla, F. Baillot, Combust. Flame 114 (3–4) (1998) 303–318. [21] S.J. Shanbhogue, T.C. Lieuwen, Response of a Rod Stabilized, Premixed Flame to Longitudinal Acoustic Forcing, GT2006-90302, Presented at Proceedings of the ASME Turbo Expo, Barcelona, 2006. [22] B. Be´dat, R.K. Cheng, Combust. Flame 100 (3) (1995) 485–494. [23] C.K. Law, C.J. Sung, Prog. Energy Combust. Sci. 26 (4–6) (2000) 459–505. [24] U.G. Hegde, D. Reuter, B.R. Daniel, B.T. Zinn, Combust. Sci. Technol. 55 (4) (1987) 125–138. [25] G.S. Jiang, D. Peng, SIAM J. Sci. Comput. 21 (2000) 2126–2143. [26] S. Gottlieb, C.W. Shu, Math. Comput. 67 (221) (1998) 73–85.
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Proceedings of the Combustion Institute 32 (2009) 1787–1794
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Flame sheet dynamics of bluff-body stabilized flames during longitudinal acoustic forcing Santosh Shanbhogue *, Dong-Hyuk Shin, Santosh Hemchandra, Dmitriy Plaks, Tim Lieuwen Ben T. Zinn Combustion Laboratory, Georgia Institute of Technology, 635 Strong Steet, Atlanta, GA 30318, USA
Abstract Bluff-body stabilized flames are susceptible to combustion instabilities due to interactions between acoustics, vortical disturbances, and the flame. In order to elucidate these flow-flame interactions during an instability, an experimental and computational investigation of the flame-sheet dynamics of a harmonically excited flame was performed. It is shown that the flame dynamics are controlled by three key processes: excitation of shear layer instabilities by the axially oscillating flow, anchoring of the flame at the bluff body, and the kinematic response of the flame to this forcing. The near-field flame features are controlled by flame anchoring and the far-field by kinematic restoration. In the near-field, the flame response grows with downstream distance due to flame anchoring, which prevents significant flame movement near the attachment point. Theory predicts that this results in linear flame response characteristics as a function of perturbation amplitude, and a monotonic growth in magnitude of the flame-sheet fluctuations near the stabilization point, consistent with the experimental data. Farther downstream, the flame response reaches a maximum and then decays due to the dissipation of the vortical disturbances and action of flame propagation normal to itself, which acts to smooth out the wrinkles generated by the harmonic flow forcing. This behavior is strongly non-linear, resulting in significant variation in far-field flame-sheet response with perturbation amplitude. Ó 2009 The Combustion Institute. Published by Elsevier Inc. All rights reserved. Keywords: Combustion instabilities; Bluff-body flames; Flame dynamics; Flame kinematics
1. Introduction The objective of this paper is to describe the flame dynamics of acoustically forced, bluff-body flames. The motivation for this work is 2-fold.
* Corresponding author. Address: Reacting Gas Dynamics Laboratory, 3-339 Mechanical Engineering, 77 Massachusetts Avenue, Massachusetts Institute of Technology, Cambridge, MA 02139, USA. Fax: +1 617 253 5981. E-mail address: [email protected] (S.J. Shanbhogue).
First, the dynamics of such flames, even in the absence of imposed excitation, involves complex interactions between gas expansion effects, flow instabilities, and vorticity dynamics induced by both shear and combustion, as highlighted by, e.g., Poinsot and Veynante [1], Schadow et al. [2] and Rogers and Marble [3]. As such, this is a fundamental problem that involves complex interactions between a variety of fluid mechanic and combustion processes [4,5]. In addition, it is motivated by the problem of combustion instabilities in devices utilizing these types of flame-holders, such as afterburners [3]. In many such instances,
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it is known that vortical structures generated from these flame-holders interact with the flame downstream, causing its heat release to oscillate [6]. In the non-reacting case, the flow-field dynamics are dominated by that of the free shear layer and wake, each exhibiting distinct instabilities with their respective frequency/amplitude response characteristics [7]. For the conditions explored in this paper, the instability of the free shear layer is of particular significance [3,8]. The shear layer (or Kelvin–Helmholtz, KH) instability is a convective instability leading to vortex roll-up and pairing. Under the influence of harmonic excitation, the separated shear layer rolls up into vortices with a frequency equal to that of the imposed excitation [9]. In addition, they lead to velocity fluctuations observed at sum and difference frequencies of the forcing frequency and its harmonics, due to non-linear interactions [9]. In the presence of heat release, the flow dynamics are fundamentally altered. For example, studies of the impact of heat release on planar shear layers indicate reduced vortex structure growth rates and Reynolds stresses compared to the non-reacting situation [10,11]. For premixed bluff-body flames, the most prominent effect is the suppression of the absolute wake instability, which leads to the von-Ka´rma´n vortex street, for flames with density ratios greater than 2 [12]. This is the reason that the shear layer instability dominates the flame response characteristics highlighted in this paper. A number of prior studies have characterized the interaction of flames with harmonic waves arising due to both acoustic waves [13] and convecting, vortical disturbances [14–18]. The dynamics of the flame are controlled by flame kinematics, i.e., the propagation of the flame normal to itself at the local burning velocity, and the flow-field that the flame is locally propagating into. This is mathematically described by the G-equation [1,19]: oG þ~ u rG ¼ S L jrGj ot
ð1Þ
where the flame position is described by the parametric equation Gð~ x; tÞ ¼ 0. Also, ~ u ¼~ uð~ x; tÞ and SL denote the flow-field just upstream of the flame and laminar burning velocity, respectively. In the unsteady case, the flame is being continually wrinkled by the unsteady flow-field, ~ u0 . The action of flame propagation normal to itself, the term on the right side of Eq. (1), is to smooth these wrinkles out through ‘‘Huygens propagation”/‘‘kinematic restoration”. As such, a wrinkle created at one point of the flame due to a velocity perturbation is convected downstream and diminishes in size due to kinematic restoration. Indeed, the interaction between the excitation (acoustic/vortical flow oscillations) and the damping (restoration property of the flame) can lead to a range of effects depending upon flame stabilization and the
relative values of the flow oscillations and flame speed. This manifests itself through both local influences upon the flame topology [20] (e.g., cusping, amplitude of corrugation, pocket formation), and global influences upon the overall unsteady heat release response of the flame. The goal of the present work is to elucidate the flame-sheet dynamics that results from an acoustically forced bluff-body flame. First, we experimentally study the dynamics of the perturbed bluff-body flame, i.e., the spectral and spatial characteristics of the flame position and the effects of excitation amplitude and frequency. Then, we show that the key characteristics of these response curves can be understood from an analysis of the G-equation. These theoretical solutions show that the near-field features are controlled by flame anchoring and the far-field by kinematic restoration. 2. Experimental facility and approach Experiments were carried out in a 915 mm long atmospheric pressure burner with a square crosssection (95 mm 95 mm), shown in Fig. 1a. Fuel, air and fine (1–5 lm) olive oil are introduced in a mixing chamber located at the base of the burner. The mixture exits the mixing chamber into a 150 mm tube having the same cross-section as the burner, which houses two 100 W acoustic loudspeakers. The mixture then passes through an aluminum honeycomb flow straightening section. The bluff body is mounted at the immediate exit of the channel. The flame is visualized with a laser sheet that scatters light off the olive oil droplets in the reactants. A 1 Watt Coherent Innova 70-5 Ar+ laser was used for illumination. A cylindrical lens arrangement was setup to provide a 2 mm light sheet that passes through the flame at the mid point of the bluff body. The resulting field of view spanned 109 mm 109 mm and was imaged using a Phantom Ultracam3 intensified camera with a 514 nm laser line filter and a BG-7 Schott Glass absorption filter placed in front of a Nikon 55 mm, f#-2.8 lens. At each operating point and/ or excitation amplitude, 2048–4096 images were obtained at 500–1000 frames/s with an exposure time of 400 ls. The acoustic drivers were sinusoidally forced with a function generator, connected to an amplifier. Typical axial and transverse cuts of the flame sheet are shown in Fig. 1b at an amplitude of excitation where the transverse flame location is multi-valued. The vertical image is in the direction of the mean flow and the horizontal image is a transverse cut obtained by rotating the laser sheet by 90°. Velocity measurements were obtained using phase locked particle image velocimetry (PIV) using a dual head Nd:YAG laser of wavelength 532 nm and peak power output of 120 mJ/pulse.
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Fig. 1. (a) Schematic of the experimental facility; dimensions are in millimeters. (b) Instantaneous axial (obtained at bluff-body centerpoint) and span-wise (obtained at axial location y/D = 3) Mie scattering images (conditions: Uo = 1.8 m/s, fo = 130 Hz, u0 /Uo = 0.06, / = 0.8, circular bluff body).
A 1600 1200 pixels CCD camera, with an Fmount Nikon 55 mm micro-lens with an aperture of f/5.6 was used for imaging. The interval between the beam pulses was set to 30 ls. The light sheet was generated using two cylindrical lenses of 150 and 1000 mm focal length that reduced the thickness of the beam and caused the 5 mm laser beam to diverge to a height of 40 mm. The distance between the imaging plane and the camera was set at 30.5 cm. Seed consisted of 1.5 lm Al2O3 injected into the setup with a cyclone seeder. At each phase, 128 images were recorded and the resulting data was ensemble averaged to provide velocity uncertainties of approximately 2%. The field of view was split into interrogation regions of 64 64 pixels, with a 50% overlap, yielding a spatial resolution of 1.5 mm. Mean velocities (Uo) and acoustic velocity amplitudes, u0a , reported below were obtained with a calibrated hotwire anemometer at the jet exit without the bluff body mounted. Transverse mapping of the velocity field at the tube exit showed that the mean velocity and incident acoustic wave profile was planar to within 3% and 4%, respectively. Because of excitation of vorticity at the bluff body, the actual velocity disturbance exciting the flame is different from u0a . All experiments were conducted with natural gas as fuel. Triangular and circular bluff bodies with various widths, d, were used for flame stabilization. 3. Results A comparison of the flame and flow-field dynamics during forced and unforced conditions
is shown in Fig. 2, which overlays instantaneous and ensemble averaged vorticity contours with instantaneous flame-sheet locations. In the absence of acoustic forcing (Fig. 2a), i.e., when u0a =U o ¼ 0, no periodic activity is present in the flame-sheet fluctuations, as determined from spectral analysis of the flame-sheet position [21]. On an instantaneous basis, random, roughly symmetric corrugations on the flame are evident leading to a flame brush growing in thickness with downstream distance, similar to the observations of Be´dat and Cheng [22] and Knaus and Gouldin [4]. As acoustic forcing is introduced (Fig. 2b–d) the presences of symmetric distortions are evident on the flame, whose amplitudes are dependent on the excitation magnitude. At larger distances from the bluff body, particularly at higher amplitudes, cusping of the flame front can also be seen, indicative of the action of the non-linear kinematic restoration process due to flame propagation normal to itself, see Law and Sung [23]. At even higher amplitudes (Fig. 2d) the complete roll-up of the flame by the convecting vortical structure is also seen. The period of formation of these structures is identical to that of the acoustic excitation, fo. The vortical structures originate in the shear layer of the bluff body and propagate in the direction of the flow. These images also show that the vorticity/velocity field that perturbs the flame decays relatively quickly, and that the flame-sheet corrugations persist further downstream. These observations can be quantified by tracking the flame edge. The coordinate system (see Fig. 2b) is chosen such that the y-axis points in the direction of the mean flow. The (x, y) = (0, 0) points is on the downstream edge of the bluff body, on the centerline. The flame
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Fig. 2. Instantaneous images of the flame-sheet and vorticity field (a) without excitation and (b–d) with increasing amplitudes of excitation (d = 9.52 mm, Uo = 2.7 m/s, fo = 300 Hz and / = 0.72).
location, L(y,t) is determined at each time instant, t, and each spatial location, y for conditions where the flame sheet is a single valued function of y (i.e., lower u0a values, no quantitative data is shown here for flames with multi-valued L(y,t)). The mean value of L at each y location is subtracted to obtain the flame-front perturbation value L0 ðy; tÞ ¼ Lðy; tÞ LðyÞ. This signal is then decomposed into its spectral components using the Fourier transform to yield L0 (y,f).
Typical characteristics of flame-front position spectra (amplitude and phase), under the influence of acoustic excitation are shown in Fig. 3. The convective wavelength of the flame-front disturbances, kcf = Uo/fo, equals the distance a disturbance propagating at the mean flow velocity travels in one acoustic period. The plot on the left shows the spectrum of the flame response at seven downstream locations. The envelope of the flame response at f = fo is also drawn. Close to the bluff
Fig. 3. (a) Dependence of flame-sheet fluctuation spectrum, L0 (y,f) upon axial location (Uo = 4.5 m/s, fo = 300 Hz). (b) Phase dependence upon normalized axial location, where yo indicates first axial location where data was obtained (d = 9.52 mm, u0a =U o ¼ 0:05 and / = 0.67, triangular bluff body).
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body, the flame responds chiefly at the frequency of excitation (fo). Moving downstream, the response at f = fo first grows, reaches a maximum, and then decreases. This behavior is due to the growth and decay of the underlying flow structures as well as the propagation of the flame, which tends to smooth out the wrinkles. These results are consistent with other data, e.g., Hegde et al. [24], though they have apparently not been explicitly discussed. The spectrum also exhibits a monotonic increase in broadband fluctuations with downstream distance. This reflects the random flapping of the flame brush, which increases in magnitude with downstream axial distance, as is clearly seen for the unforced case, see Fig. 2a. The corresponding axial variation of the phase at the excitation frequency is plotted in Fig. 3b. For all these conditions, the phase behavior nearly collapses onto a single line when axial distance is normalized by kc, indicating a constant axial convection speed of these flamesheet disturbances of Ucf = 0.98Uo. In general, there is a mild dependence of the convection speed on the equivalence ratio and the bluff-body shape [21], ranging between 0.8 < Ucf/Uo < 1.1. Although not shown, this convection speed is nearly independent of excitation amplitudes, at least for amplitudes below where the flame becomes multi-valued [21]. Consider the amplitude characteristics in more detail. Figure 4a plots the dependence of the amplitude upon normalized axial distance at three excitation amplitudes, but all other conditions remaining equal. Figure 4b plots these same data, with the y-axis normalized by the acoustic excitation amplitude. The plot can be divided into two regions manifested by ‘‘growth” and ‘‘decay” of the flame-sheet oscillations. This plot shows that the flame response increases linearly with excitation amplitude in the bluff-body near-field, or ‘‘growth” region, as manifested by all the curves
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in Fig. 4b converging onto a single line. However, farther downstream, the flame dynamics are clearly non-linear, as again evidenced by the amplitude normalized response plots diverging with increasing axial distance. In particular, the saturation in flame response can be seen, indicating that the maximum in flame response, as well as response farther downstream, does not grow proportionally with amplitude. As shown next, this is due to the non-linearities associated with the kinematic restoration process, which plays the dominant role in this region. In contrast, in the growth region, the flame dynamics are linear and primarily influenced by the flame-anchoring condition near the stabilization point. In order to better understand these features, the next section describes computational analyses of this configuration. 4. Kinematic modeling of flame-sheet dynamics In this section, we consider the role of flame kinematic processes in the basic flame response characteristics described above. As will be shown, the flame dynamics are controlled by three processes: (1) the anchoring of the flame at the bluff body, (2) the excitation of wrinkles by the oscillating velocity, and (3) flame propagation normal to itself at the local flame speed, smoothing out the wrinkles introduced by the oscillating flow. Utilizing the coordinate system defined in Fig. 2b, we write the function G in Eq. (1) as G(x,y,t) = 0 = L(y,t) x, as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi oL oL oL v þ u ¼ SL 1 þ : ð2Þ ot oy oy This equation is solved with a flame-anchoring boundary condition at y = 0; i.e., whether and to what extent the flame base moves in response to excitation. Analytical solutions of
Fig. 4. (a) Measured dependence of flame sheet amplitude response upon normalized axial distance, kc = Uo/f. (b) Same curve, but amplitude normalized by acoustic velocity amplitude: () u0a ¼ 0:028, (h) u0a ¼ 0:021, (+) u0a ¼ 0:016, (/) u0a ¼ 0:010. Other conditions: Uo = 2.27 m/s, D = 12.7 mm, fo = 150 Hz, cylindrical bluff body.
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the linearized version of this equation are possible for low amplitudes and/or small y/kc values, but it requires numerical solutions for the general non-linear case. A fifth-order Weighted Essentially Non-Oscillatory (WENO) numerical scheme [25] was used, uniformly fifth-order accurate in space in the smooth regions and thirdorder accurate in discontinuous regions. Derivatives at the boundary nodes were calculated using third-order accurate upwind-differencing schemes so that only the nodes inside the computational domain are utilized. A third-order accurate Total Variation Diminishing (TVD) Runge– Kutta scheme [26] is used for time integration. The computations were performed for a grid of length 6kc with a grid point spacing Dy = 0.001kc. Sensitivity studies performed with a grid 10 times finer demonstrated that the difference between the two grid density results was less than 0.1%. The experimental data, e.g., Fig. 3a shows that the flame response first grows with downstream distance, up to an axial location of y/kc 0.9, before decaying. This initial increase in L0 with y is due to the flame anchoring; i.e., regardless of the perturbation, the flame attachment point remains largely fixed. As such, the amplitude of L0 must start from zero or near zero. Obviously, it will not remain zero at y > 0, since it is being perturbed by the fluctuating velocity field— hence the initial increase in L0 with y. Even if the flame attachment point vibrates some, i.e., L0 (y = 0,t) – 0, as long as the flame base does not move in phase and with the same amplitude as the flow-field, similar behavior occurs. Furthermore, because the amplitude of fluctuations in the y 0 region are so small, non-linear effects are negligible.rThese non-linear effects are contained ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi in the S L 1 þ oL term, which describes flame oy propagation normal to itself. As such, the flame dynamics near the attachment point are described by a linearized equation form of Eq. (2). An explicit solution for the slope of the jL0 (y,fo)j vs. y curve can be derived from Eq. (2) near the attachment point: ojL0 ðy ¼ 0; fo Þj ju0n ðy ¼ 0; fo Þj 1 ¼ oy U t ðy ¼ 0Þ cos2 h
ð3Þ
where u0n ; U t , and h denote the fluctuating velocity normal to the flame, mean velocity component tangential to the flame, and mean flame angle with respect the axial coordinate, see Fig. 2a. This equation assumes that L0 (y = 0,t) = 0 at y = 0. It can be seen that the near-field flame response slope increases linearly with perturbation magnitude, u0n . This result, together with the result in Fig. 4 suggests that the near-field flame response is controlled purely by the velocity field near the flame attachment point. This explains why the
curves in Fig. 4 converge to a common behavior at y 0. The peaking and subsequent reduction in amplitude of flame response shown in Fig. 4 is the result of flame propagation normal to itself which destroys flame wrinkles—a non-linear effect [23]. Moreover, for a thermo-diffusively stable flame, unsteady curvature effects also work to destroy flame wrinkles—even at first-order in perturbation amplitude—due to the increase in flame speed at locations concave to the flow and vice versa [18]. If the perturbation velocity persisted indefinitely downstream, at some point the flame wrinkle excitation and destruction processes would equilibrate, leading to a constant or oscillating amplitude of flame perturbation with downstream distance. However, the amplitude of the excitation field decays with downstream distance, as shown in Fig. 2. Hence, the amplitude of flame wrinkling decays. PIV velocity field measurements were obtained at one test condition to enable a comparison between predictions and calculations. Based upon the data, the transverse velocity field at the flame was fit to an equation of the form: vðy; tÞ ¼ vo ðyÞ þ ev y cosðxðt y=U c ÞÞecy :
ð4Þ
This equation shows that the perturbation is propagating axially with a velocity Uc and decaying at a rate given by c. The vortex convection velocity, Uc, is not necessarily equal to the flame-sheet convection velocity, Ucf, discussed in the context of Fig. 3. A similar expression, motivated by the continuity equation, was used for the axial velocity component, u0 (y,t) = e(1 + cy) ecycos(x(t y/Uc) w). These velocity field parameters were fit from the data, leading to c = 2.28/kc, w = 2p/3, Uc/Uo = 0.8, e = 1.4ev/c2kc (same conditions as in Fig. 3). A comparison of these velocity fits and the measurements are shown in Fig. 5. In addition, the frequency, fo, was determined from the known forcing frequency, h was determined from the flame images in the unforced case, and L0 (y = 0,t) was estimated as zero, as was observed in measurements (Fig. 4)—these quantities are also needed to solve Eq. (2). Direct comparisons of flame position dynamics were made in the near-field region only for two reasons. First, obtaining good velocity field data requires excitation at amplitudes that leads to the flame becoming multi-valued (e.g., see Fig. 2d) downstream. A further complication, particularly with velocity field specification further downstream, occurs because of flame-sheet movement—measurements of the ensemble average perturbation velocity field just upstream of the flame front (i.e., at a temporally varying location), hu(x(t),y(t))i, or at a spatially fixed point coinciding with some ‘‘average” flame-front location, hu(xo,yo)i, give different results. Both results
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Fig. 5. Comparison between the model and measurements of perturbation velocity amplitudes upstream of flame surface. Varying point: velocity measured at a location which is fixed relative to the moving flame. Fixed point: velocity measured at a fixed point in space. Conditions same as in Fig. 3.
are illustrated in Fig. 5 (compare ‘‘Fixed point” and ‘‘Varying point”). Nonetheless, very good quantitative comparisons are possible in the flame near-field, see Fig. 6. Starting first with the amplitude, inserting the measured velocity field data into Eq. (3), leads 0 o Þj 1 to a predicted near-field slope of junUðy;f ¼ cos2 h t ðyÞ 0:15 0:04=0:02 (using hu(x(t),y(t))i), in excellent 0 o Þj ¼ 0:16 agreement with the value of ojL ðy¼0;f oy 0:01 that was estimated from the flame-sheet fluctuation measurements. A corresponding comparison of the phase is also shown in Fig. 6—notice that their slopes are nearly identical, differing by about 2%. The two curves have a nearly constant offset for larger y values, due to a variation in the experimental phase slope near the attachment point, y ? 0. Computed solutions for several amplitudes and over a larger range of axial positions of flame
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Fig. 7. Predicted flame-sheet response based upon solution of non-linear flame-front equation at various perturbation amplitudes. For all cases c = 2.28/kc, w = 2p/3, Uc/Uo = 0.8, e = 1.4ev/c2kc. Other conditions same as in Fig. 3.
position amplitude are shown in Fig. 7. Notice that the behavior observed experimentally (Fig. 4a) is also seen here. First, the flame response is characterized by a near-field growth and a far-field decay region. Next, the flame response increases linearly with axial distance in the near-field region. Although not plotted, normalization of these curves by perturbation amplitude collapses them onto a single line, similar to the data in Fig. 4 (right), demonstrating the linearity in near-field flame response with amplitude as well. Next, Fig. 7 shows the divergence of these curves in the far-field region, demonstrating the far-field non-linear response. Finally, it captures the increased destruction rate of flame wrinkling, as manifested by the increased negative slope in the ‘‘decay” region, with perturbation amplitude. This latter point is illustrated by the growing difference between the linearized solution (scaled by perturbation amplitude) and the full non-linear solution with increasing y. As such, this shows the far-field decay characteristics of the flame surface amplitude are due to both velocity field decay and flame kinematic non-linearity.
5. Concluding remarks
Fig. 6. Comparison of axial dependence of flame-sheet amplitude and phase between experiments and theory. Conditions same as Fig. 3.
In this paper, two primary contributions have been made in understanding the response of bluff-body flames to harmonic excitation. First, through experiments, the coherent flame response was shown to have two different behaviors in the near and far-field of the bluff body, dominated by linear and non-linear flame kinematic processes, respectively. Second, using a kinematic model that encapsulated the dominant features of the perturbing velocity field; the key processes controlling the response have been
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identified as (1) the anchoring of the flame at the bluff body, (2) the excitation of flame-front wrinkles by the oscillating velocity, and (3) flame propagation normal to itself at the local flame speed. The first two processes control the growth of the flame response and the last process controls its decay. Acknowledgments This research was supported by the US-DoE and NSF under contracts DE-FG26-07NT43069 and CBET-0651045; contract monitors Rondle Harp and Dr. Phil Westmoreland, respectively. References [1] T. Poinsot, D. Veynante, Theoretical and Numerical Combustion, RT Edwards Inc., Flourtown, PA, 2001. [2] K.C. Schadow, E. Gutmark, T.P. Parr, D.M. Parr, K.J. Wilson, J.E. Crump, Combust. Sci. Technol. 64 (4) (1989) 167–186. [3] D.E. Rogers, F.E. Marble, Jet Propulsion 26 (1) (1956) 456–462. [4] D.A. Knaus, F.C. Gouldin, Proc. Combust. Inst. 28 (2000) 367–373. [5] F. Baillot, D. Durox, S. Ducruix, G. Searby, L. Boyer, Combust. Sci. Technol. 142 (1) (1999) 91– 109. [6] D.M. Kang, F.E.C. Culick, A. Ratner, Combust. Flame 151 (3) (2007) 412–425. [7] C.H.K. Williamson, Annu. Rev. Fluid Mech. 28 (1996) 477–539. [8] P.L. Blackshear, W.D. Rayle, L.K. Tower, Study of Screeching Combustion in a 6-in. Simulated Afterburner, Report No. NACA TN 3567, 1955. [9] A. Kourta, H.C. Boisson, P. Chassaing, H.H. Minh, J. Fluid Mech. 181 (1987) 141–161.
[10] P.A. Mcmurtry, J.J. Riley, R.W. Metcalfe, J. Fluid Mech. 199 (1989) 297–332. [11] J.C. Hermanson, P.E. Dimotakis, J. Fluid Mech. 199 (1989) 333–375. [12] R.R. Erickson, M.C. Soteriou, P.G. Mehta, The Influence of Temperature Ratio on the Dynamics of Bluff Body Stabilized Flames, AIAA Paper No. 2006-753, Presented at 44th AIAA Aerospace Sciences Meeting and Exhibit, Reno, Nevada, 2006. [13] T. Lieuwen, Proc. Combust. Inst. 30 (2) (2005) 1725–1732. [14] A. Chaparro, E. Landry, B.M. Cetegen, Combust. Flame 145 (1–2) (2006) 290–299. [15] R. Balachandran, B.O. Ayoola, C.F. Kaminski, A.P. Dowling, E. Mastorakos, Combust. Flame 143 (1–2) (2005) 37–55. [16] C. Nottin, R. Knikker, M. Boger, D. Veynante, Proc. Combust. Inst. 28 (2000) 67–73. [17] T. Schuller, S. Ducruix, D. Durox, S. Candel, Proc. Combust. Inst. 29 (1) (2002) 107–113. [18] Preetham, T. Lieuwen, Nonlinear Flame-Flow Transfer Function Calculations: Flow Disturbance Celerity Effects, AIAA Paper No. 2004-4035, Presented at the 40th AIAA/ASME/SAE Joint Propulsion Conference, 2004. [19] V. Yang, F.E.C. Culick, Combust. Sci. Technol. 45 (1) (1986) 1–25. [20] A. Bourehla, F. Baillot, Combust. Flame 114 (3–4) (1998) 303–318. [21] S.J. Shanbhogue, T.C. Lieuwen, Response of a Rod Stabilized, Premixed Flame to Longitudinal Acoustic Forcing, GT2006-90302, Presented at Proceedings of the ASME Turbo Expo, Barcelona, 2006. [22] B. Be´dat, R.K. Cheng, Combust. Flame 100 (3) (1995) 485–494. [23] C.K. Law, C.J. Sung, Prog. Energy Combust. Sci. 26 (4–6) (2000) 459–505. [24] U.G. Hegde, D. Reuter, B.R. Daniel, B.T. Zinn, Combust. Sci. Technol. 55 (4) (1987) 125–138. [25] G.S. Jiang, D. Peng, SIAM J. Sci. Comput. 21 (2000) 2126–2143. [26] S. Gottlieb, C.W. Shu, Math. Comput. 67 (221) (1998) 73–85.